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|
%D \module
%D [ file=m-matrix,
%D version=2014.11.04, % already a year older
%D title=\CONTEXT\ Extra Modules,
%D subtitle=Matrices,
%D author={Jeong Dalyoung \& Hans Hagen},
%D date=\currentdate,
%D copyright={PRAGMA ADE \& \CONTEXT\ Development Team}]
%C
%C This module is part of the \CONTEXT\ macro||package and is
%C therefore copyrighted by \PRAGMA. See mreadme.pdf for
%C details.
%D This code is based on a post by Dalyoung on the context list. After that
%D we turned it into a module and improved the code a bit. Feel free to ask
%D us for more. Once we're satisfied, a more general helper l-matrix could
%D be made. Dalyoung does the clever bits, and Hans only cleanes up and
%D optimizes a bit.
% \registerctxluafile{l-matrix}{1.001} % not yet
\startmodule[matrix]
\startluacode
local settings_to_hash = utilities.parsers.settings_to_hash
local formatters = string.formatters
local copy = table.copy
local insert = table.insert
local remove = table.remove
local random = math.random
local context = context
local matrix = { }
moduledata.matrix = matrix
local f_matrix_slot = formatters["%s_{%s%s}"]
function matrix.symbolic(sym, x, y, nx ,ny) -- symMatrix("a", "m", "n")
local nx = nx or 2
local ny = ny or nx
local function filled(i,y)
local mrow = { }
for j=1,nx do
mrow[#mrow+1] = f_matrix_slot(sym,i,j)
end
mrow[#mrow+1] = "\\cdots"
mrow[#mrow+1] = f_matrix_slot(sym,i,y)
return mrow
end
local function dummy()
local mrow = { }
for j=1,nx do
mrow[#mrow+1] = "\\vdots"
end
mrow[#mrow+1] = "\\ddots"
mrow[#mrow+1] = "\\vdots"
return mrow
end
--
local mm = { }
for i=1,ny do
mm[i] = filled(i,y)
end
mm[#mm+1] = dummy()
mm[#mm+1] = filled(x,y)
return mm
end
-- todo: define a matrix at the tex end so that we have more control
local fences = {
parentheses = { left = "\\left(\\,", right = "\\,\\right)" },
brackets = { left = "\\left[\\,", right = "\\,\\right]" },
bars = { left = "\\left|\\,", right = "\\,\\right|" },
}
-- one can add more fences
fences.bar = fences.bars
fences.parenthesis = fences.parentheses
fences.bracket = fences.brackets
-- one can set the template
matrix.template = "%0.3F"
function matrix.typeset(m,options)
if type(m) == "table" then
local options = settings_to_hash(options or "")
local whatever = options.determinant == "yes" and fences.bars or fences.parentheses
if options.fences then
whatever = fences[options.fences] or whatever
elseif options.determinant then
-- whatever = fences.brackets
whatever = fences.bars
end
local template = options.template or matrix.template
if template == "yes" then
template = matrix.template
elseif template == "no" then
template = false
elseif tonumber(template) then
template = "%0." .. template .. "F"
end
context.startmatrix(whatever)
for i=1, #m do
local mi = m[i]
for j=1,#mi do
context.NC()
local n = mi[j]
if template and tonumber(n) then
context(template,n)
else
context(mi[j])
end
end
context.NR()
end
context.stopmatrix()
elseif m then
context(m)
end
end
-- interchange two rows (i-th, j-th)
-- function matrix.swaprows(t,i,j)
-- if i <= #t and j <= #t then
-- t[i], t[j] = t[j], t[i]
-- return t
-- else
-- return "error: out of bound"
-- end
-- end
function matrix.swaprows(t,i,j)
local ti = t[i]
if not ti then
return "error: no row i"
end
local tj = t[j]
if not tj then
return "error: no row j"
end
t[i], t[j] = tj, ti
return t
end
-- interchange two columns (i-th, j-th)
-- function matrix.swapcolumns(t, i, j)
-- if i <= #t[1] and j <= #t[1] then
-- for k = 1, #t do
-- t[k][i], t[k][j] = t[k][j], t[k][i]
-- end
-- return t
-- else
-- return "error: out of bound"
-- end
-- end
function matrix.swapcolumns(t, i, j)
local t1 = t[1]
if not t1 then
return "error: no rows"
end
local n = #t1
if i <= n then
return "error: no row i"
end
if j <= n then
return "error: no row j"
end
for k = 1, #t do
local tk = t[k]
tk[i], tk[j] = tk[j], tk[i]
end
return t
end
matrix.swapC = matrix.swapcolumns
matrix.swapR = matrix.swaprows
matrix.swap = matrix.swaprows
-- replace i-th row with factor * (i-th row)
function matrix.multiply(m,i,factor)
local mi = m[i]
for k=1,#mi do
mi[k] = factor * mi[k]
end
return m
end
-- scalar product "factor * m"
function matrix.scalar(m, factor)
for i=1,#m do
local mi = m[i]
for j=1,#mi do
mi[j] = factor * mi[j]
end
end
return m
end
-- replace i-th row with i-th row + factor * (j-th row)
function matrix.sumrow(m,i,j,factor)
local mi = m[i]
local mj = m[j]
for k=1,#mi do
mi[k] = mi[k] + factor * mj[k]
end
end
-- transpose of a matrix
function matrix.transpose(m)
local t = { }
for j=1,#m[1] do
local r = { }
for i=1,#m do
r[i] = m[i][j]
end
t[j] = r
end
return t
end
-- inner product of two vectors
function matrix.inner(u,v)
local nu = #u
if nu == 0 then
return 0
end
local nv = #v
if nv ~= nu then
return "error: size mismatch"
end
local result = 0
for i=1,nu do
result = result + u[i] * v[i]
end
return result
end
-- product of two matrices
function matrix.product(m1,m2)
if #m1[1] == #m2 then
local product = { }
for i=1,#m1 do
local m1i = m1[i]
local mrow = { }
for j=1,#m2[1] do
local temp = 0
for k=1,#m1[1] do
temp = temp + m1i[k] * m2[k][j]
end
mrow[j] = temp
end
product[i] = mrow
end
return product
else
return "error: size mismatch"
end
end
local function uppertri(m,sign)
local temp = copy(m)
for i=1,#temp-1 do
local pivot = temp[i][i]
if pivot == 0 then
local pRow = i +1
while temp[pRow][i] == 0 do
pRow = pRow + 1
if pRow > #temp then -- if there is no nonzero number
return temp
end
end
temp[i], temp[pRow] = temp[pRow], temp[i]
if sign then
sign = -sign
end
end
local mi = temp[i]
for k=i+1, #temp do
local factor = -temp[k][i]/mi[i]
local mk = temp[k]
for l=i,#mk do
mk[l] = mk[l] + factor * mi[l]
end
end
end
if sign then
return temp, sign
else
return temp
end
end
matrix.uppertri = uppertri
local function determinant(m)
if #m == #m[1] then
local d = 1
local t, s = uppertri(m,1)
for i=1,#t do
d = d * t[i][i]
end
return s*d
else
return "error: not a square matrix"
end
end
matrix.determinant = determinant
local function rowechelon(m,r)
local temp = copy(m)
local pRow = 1
local pCol = 1
while pRow <= #temp do
local pivot = temp[pRow][pCol]
if pivot == 0 then
local i = pRow
local n = #temp
while temp[i][pCol] == 0 do
i = i + 1
if i > n then
-- no nonzero number in a column
pCol = pCol + 1
if pCol > #temp[pRow] then
-- there is no nonzero number in a row
return temp
end
i = pRow
end
end
temp[pRow], temp[i] = temp[i], temp[pRow]
end
local row = temp[pRow]
pivot = row[pCol]
for l=pCol,#row do
row[l] = row[l]/pivot
end
if r == 1 then
-- make the "reduced row echelon form"
local row = temp[pRow]
for k=1,pRow-1 do
local current = temp[k]
local factor = -current[pCol]
local mk = current
for l=pCol,#mk do
mk[l] = mk[l] + factor * row[l]
end
end
end
-- just make the row echelon form
local row = temp[pRow]
for k=pRow+1, #temp do
local current = temp[k]
local factor = -current[pCol]
local mk = current
for l=pCol,#mk do
mk[l] = mk[l] + factor * row[l]
end
end
pRow = pRow + 1
pCol = pCol + 1
if pRow > #temp or pCol > #temp[1] then
pRow = #temp + 1
end
end
return temp
end
matrix.rowechelon = rowechelon
matrix.rowEchelon = rowechelon
-- make matrices until its determinant is not 0
function matrix.make(n,m,low,high)
if not n then
n = 10
end
if not m then
m = 10
end
if not low then
low = 0
end
if not high then
high = 100
end
local t = { } -- make an empty n1 x n2 array
local again = true
for i=1,n do
t[i] = { }
end
while true do
for i=1,n do
local ti = t[i]
for j=1,m do
ti[j] = random(low,high)
end
end
if n ~= m or determinant(t,1) ~= 0 then
return t
end
end
end
-- extract submatrix by using
local function submatrix(t,i,j)
local rows = #t
local columns = #t[1]
local sign = 1 -- not used
if i <= rows and j <= columns then
local c = copy(t)
remove(c,i)
for k=1,rows-1 do
remove(c[k],j)
end
return c
else
return "error: out of bound"
end
end
matrix.submatrix = submatrix
-- calculating determinant using Laplace Expansion
function matrix.laplace(t) -- not sure if this is the most effient but
local factors = { 1 } -- it's not used for number crunching anyway
local data = copy(t)
local det = 0
while #data > 0 do
local mat = { }
local siz = #data[1]
if siz == 0 then
return "error: no determinant"
elseif siz == 1 then
det = data[1][1]
return det
end
for i=1,siz do
mat[i] = data[1]
remove(data,1)
end
local factor = remove(factors,1)
local m1 = mat[1]
if siz == 2 then
local m2 = mat[2]
det = det + factor * (m1[1]*m2[2] - m1[2]*m2[1])
else
for j=1,#m1 do
local m1j = m1[j]
if m1j ~= 0 then
insert(factors, (-1)^(j+1) * factor * m1j)
local m = submatrix(mat,1,j)
for k, v in next, m do
insert(data,v)
end
end
end
end
end
return det
end
-- solve the linear equation m X = c
local function solve(m,c)
local n = #m
if n ~= #c then
return copy(m)
end
local newm = copy(m)
local temp = copy(c)
for i=1,n do
insert(newm[i],temp[i])
end
return rowechelon(newm,1)
end
matrix.solve = solve
-- find the inverse matrix of m
local function inverse(m)
local n = #m
local temp = copy(m)
if n ~= #m[1] then
return temp
end
for i=1,n do
for j=1,n do
insert(temp[i],j == i and 1 or 0)
end
end
temp = rowechelon(temp,1)
for i=1,n do
for j=1,n do
remove(temp[i], 1)
end
end
return temp
end
matrix.inverse = inverse
\stopluacode
\stopmodule
\unexpanded\def\ctxmodulematrix#1{\ctxlua{moduledata.matrix.#1}}
\continueifinputfile{m-matrix.mkiv}
\starttext
\startluacode
document.DemoMatrixA = {
{ 0, 2, 4, -4, 1 },
{ 0, 0, 2, 3, 4 },
{ 2, 2, -6, 2, 4 },
{ 2, 0, -6, 9, 7 },
{ 2, 3, 4, 5, 6 },
{ 6, 6, -6, 6, 6 },
}
document.DemoMatrixB = {
{ 0, 2, 4, -4, 1 },
{ 0, 0, 2, 3, 4 },
{ 2, 2, -6, 2, 4 },
{ 2, 0, -6, 9, 7 },
{ 2, 2, -6, 2, 4 },
{ 2, 2, -6, 2, 4 },
}
\stopluacode
\startsubject[title={A symbolic matrix}]
\ctxmodulematrix{typeset(moduledata.matrix.symbolic("a", "m", "n"))}
\ctxmodulematrix{typeset(moduledata.matrix.symbolic("a", "m", "n", 4, 8))}
\stopsubject
\startsubject[title={Swap two rows (2 and 4)}]
\startluacode
moduledata.matrix.typeset(document.DemoMatrixA)
context.blank()
moduledata.matrix.swap(document.DemoMatrixA, 2, 4)
context.blank()
moduledata.matrix.typeset(document.DemoMatrixA)
\stopluacode
\stopsubject
\startsubject[title={Multiply $3 \times r_2$}]
\startluacode
moduledata.matrix.typeset(document.DemoMatrixA)
context.blank()
moduledata.matrix.typeset(moduledata.matrix.multiply(document.DemoMatrixA, 2, 3))
\stopluacode
\stopsubject
\startsubject[title={Row 2 + $4 \times r_3$}]
\startluacode
moduledata.matrix.typeset(document.DemoMatrixA)
context.blank()
moduledata.matrix.sumrow(document.DemoMatrixA, 2, 3, 4)
context.blank()
moduledata.matrix.typeset(document.DemoMatrixA,{ fences = "bars" } )
\stopluacode
\stopsubject
\startsubject[title={Transpose a matrix}]
\startluacode
moduledata.matrix.typeset(document.DemoMatrixA)
context.blank()
moduledata.matrix.typeset(moduledata.matrix.transpose(document.DemoMatrixA))
\stopluacode
\stopsubject
\startsubject[title={The inner product of two vectors}]
\startluacode
context(moduledata.matrix.inner({ 1, 2, 3 }, { 3, 1, 2 }))
context.blank()
context(moduledata.matrix.inner({ 1, 2, 3 }, { 3, 1, 2, 4 }))
\stopluacode
\startsubject[title={The product of two matrices}]
\startluacode
moduledata.matrix.typeset(document.DemoMatrixA)
context.blank()
moduledata.matrix.typeset(moduledata.matrix.product(document.DemoMatrixA,document.DemoMatrixA))
\stopluacode
\stopsubject
\startsubject[title={An Upper Triangular Matrix}]
\ctxmodulematrix{typeset(moduledata.matrix.uppertri(document.DemoMatrixB))}
\startsubject[title={A determinant}]
\startluacode
local m = {
{ 1, 2, 4 },
{ 0, 0, 2 },
{ 2, 2, -6 },
}
context(moduledata.matrix.determinant(m, "determinant=yes" ))
\stopluacode
\stopsubject
\startsubject[title={Row echelon form}]
\startluacode
local m = {
{ 1, 3, -2, 0, 2, 0, 0 },
{ 2, 6, -5, -2, 4, -3, -1 },
{ 0, 0, 5, 10, 0, 15, 5 },
{ 2, 6, 0, 8, 4, 18, 6 },
}
moduledata.matrix.typeset(m)
context.blank()
moduledata.matrix.typeset(moduledata.matrix.rowechelon(m,1), { determinant = "yes" })
\stopluacode
\stopsubject
\startsubject[title={Solving linear equation}]
\startluacode
local m = {
{ 1, 3, -2, 0 },
{ 2, 0, 1, 2 },
{ 6, -5, -2, 4 },
{ -3, -1, 5, 10 },
}
local c = { 5, 2, 6, 8 }
moduledata.matrix.typeset(moduledata.matrix.solve(m,c))
context.blank()
moduledata.matrix.typeset(moduledata.matrix.solve(m,c), { template = 6 })
context.blank()
moduledata.matrix.typeset(moduledata.matrix.solve(m,c), { template = "no" })
context.blank()
moduledata.matrix.typeset(moduledata.matrix.solve(m,c), { template = "%0.3f" })
context.blank()
moduledata.matrix.typeset(moduledata.matrix.solve(m,c), { template = "%0.4F" })
\stopluacode
\stopsubject
\startsubject[title={Inverse matrix}]
\startcombination[2*1]
{\ctxlua{moduledata.matrix.typeset { { 1, 1, 1 }, { 0, 2, 3 }, { 3, 2, 1 } }}} {}
{\ctxlua{moduledata.matrix.typeset(moduledata.matrix.inverse { { 1, 1, 1 }, { 0, 2, 3 }, { 3, 2, 1 } })}} {}
\stopcombination
\stopsubject
\stoptext
|